6. Fields

This is article 6 in the sequence entitled Fields and Particles: with Math. Here is the previous article.

We’ve actually been dealing with fields for a while now, but I didn’t say so — because I wanted to focus on the waves in these fields. In describing what the waves have been doing, we’ve been expressing their shape and their dependence on time using a quantity Z(x,t). Well, Z(x,t) is a field; it is a function of space and time with an equation of motion that determines its behavior. (A suitable equation of motion will be one such that if Z increases or decreases at a particular location, it tends to make Z increase or decrease in nearby locations at a later time; this feature commonly allows waves to be among the solutions to the equation.)

In this article we’re going to look at a few examples of fields Z(x,t) whose equations of motion allow for waves. The physical interpretation of these fields will be very different; they describe different properties of different materials. But the equations that they satisfy, and the waves they exhibit, will satisfy the same (or very similar) mathematics, so they will behave very similarly despite their very different physical origins. This will turn out to be very important.

And then we’re going to do something radical. We’ll consider fields in the context of special relativity [completely avoiding the math of relativity, and just focusing on its consequences for the fields.] As Einstein showed, if you tweak space and time to make them behave differently from the way most of us would expect, you can have a new type of field — one whose physical interpretation is not that it is a property of something else, but rather that it is a fundamental physical object all on its own.

Some Ordinary Fields Describing Ordinary Stuff

A field Z(x,t) may represent many different physical quantities. Examples include

the height of a rope extended horizontally across a room

the height of water in a river

the density of a crystal or of a gas

the orientation of atoms in a magnet

the speed of the wind

the temperature, density or pressure of the air

In each of these cases there’s a field Z(x,t): a height field, a density field, an orientation field, a wind field, a temperature field. Its value as a function of space and time tells us what the height or density or orientation or wind speed or temperature of some material medium — rope, river, crystal, gas, magnet, air — is at all places and all times. Its equation of motion tells us what possible behaviors are allowed for Z(x,t); and it also tells us how to predict the future behavior of Z(x,t) if we know its present and recent past behavior.

In each example there is a field and a medium, and we must not confuse the field with the medium. The field merely describes and characterizes one of the many properties of the relevant medium. And very different media may have very similar-behaving fields, with very similar waves — as we’ll see.

Let me emphasize one last time a point that often causes confusion. In general the field may have nothing to do with a physical distance in space. Yes, in articles 3 and 4 I used the example of a wave on a rope to illustrate what Z(x,t) might be, because it’s nice and intuitive; and also I’ve made graphs of Z(x,t) for waves many times. Both of those things might tend to give you the erroneous impression that Z(x,t) always has to do with waves that cause a physical object (like the rope) to move a distance Z in a direction of space perpendicular to the x direction. But that’s not true, as three of our four examples will illustrate.

A RopeHeight Field

First, let’s give consider the wiggling rope that was our initial example for waves (see Figure 2 of article 3): in this case the role of Z(x,t) is the height field, which we’ll call H(x,t), telling us how high the rope is at each point in space x along the rope, at every time t. If the rope just sits at its equilibrium height H0, then H(x,t) = H0; the height field is constant in space at all times. If it has a simple wave moving on it, the height field will be described by our famous wave formula that we saw in previous articles. [H(x,t) can behave in other ways too, if it is impinged on by external forces. But we won’t consider such behaviors for now.]

If we know H(x,t), we know how high the rope is at all points in space and time. Alternatively, if we know what the rope is doing now and in the very recent past, we can predict, using the equation of motion, what it will do in the future. None of this tells us very much about the rope itself. The height field just tells us what its name implies: the height of the rope. The rope is the physical medium whose height is represented as the field H(x,t); this tells us nothing about the rope’s color, thickness, tension, material, etc.

In Figure 1 I’ve given you an animation that shows a wave in the height field, coming in from the left and moving off to the right. Naively it would seem I’ve plotted the same things twice, once in orange and once in green. But they’re not the same thing. The orange curve is supposed to be the rope itself, moving in physical space. The green curve is a graph, a simple mathematics plot that just represents what’s going on with H(x,t), without reference to what H(x,t) means (i.e. height) or what its medium is. In this case only, the green graph looks just like what’s happening in the physical world. But that won’t be the case for any of the other examples that we’re about to see.

Fig. 1 (click to animate): A wave in the height field of a rope causes it to oscillate up and down. Below: The orange rope itself physically moves up and down. Above: In green, a graph of the height field H(x,t), versus x, as it evolves in time t. (M. Strassler 2012; All Rights Reserved)

A LatticeDisplacement Field

Suppose we have a medium that consists of a crystal of equally spaced atoms. (I’ve drawn this with one space dimension in Figure 2; we could consider a similar situation with three space dimensions, but that would be an unnecessary complication for the moment. I’ve also labelled every 10th atom red so you can keep track of its motion more easily. And I’ve highly exaggerated the distance between the atoms; you might want to think about there being a few million atoms between each red atom rather than 10.)

Let’s look at the displacement field D(x,t) that tells us how far, at time t, the atom that is normally at equilibrium position x has moved away from its equilibrium position in the lattice. That means that in the static picture shown in Figure 2, the field is zero everywhere, D(x,t)=0, because all atoms are in their normal positions in the lattice. When you click to animate the figure, what you’ll see is (a) below, the individual atoms oscillate back and forth in a motion that, overall, propagates as a wave from left to right; (b) above, a graph of the lattice displacement field D(x,t) shows how it behaves as the wave moves past. Note D(x,t) wiggles back and forth in much the same way the height field did in Figure 1; the waves in the two fields behave similarly, even though the physical interpretation of the field is very different in the two cases.

Fig. 2 (click to animate.): A wave displaces the positions of atoms in a lattice away from their normal “equilibrium” positions. Below: The atoms oscillate back and forth as the wave passes; to make this easier to see, every 10th atom has been marked in red. Above: A graph of the displacement field D(x,t) that indicates how far each atom that is normally at the position x has been displaced; its evolution with time t is shown. (M. Strassler 2012; All Rights Reserved)

A MagnetOrientation Field

What is a permanent magnet? It consists of a set of atoms, each of which is a tiny little magnet with a tiny magnetic field, all aligned so as to create cooperatively a large magnetic field. A magnet is shown in Figure 3; each atom points vertically. In this case, the orientation field Θ(x,t) tells us how far, at time t, the atom at position x points away from the vertical; Θ, in short, is the angle between each atom’s little magnet and the vertical direction. The animation in Figure 3 shows a wave in the magnet, where orientations of the atomic magnets oscillate left and right of vertical. Above the magnet, in green, is a graph of Θ(x,t); yet again, it looks the same as in the previous cases.

Fig. 3 (click to animate): Below: A wave causes the orientation of magnetic atoms to oscillate inside a magnet. Above: In green, a graph of the orientation field Θ(x,t) as a function of x, as it evolves in time t. (M. Strassler 2012; All Rights Reserved)

An AirPressure Field

Let’s consider a gas of molecules placed in a long tube; the long direction of the tube we’ll call the direction x. The molecules in a gas move around almost randomly, banging into the walls of the tube and into each other. In equilibrium, the density (the number of molecules in a certain volume) and the pressure P(x,t) (the force per unit area that would be exerted by the molecules on the surface of a little ball inserted at the point x at time t) are constant. But sound waves passing through the gas will make the pressure (and density) oscillate, as shown in Figure 4 (click to animate). The density and pressure increase and decrease repeatedly; the molecules move back and forth, though on average they do not move at all, even though the wave and its energy move from left to right across the gas. The graph of P(x,t) looks yet again the same as before.

Fig. 4 (click to animate); A pressure wave moves through a tube containing gas molecules (blue dots.) Below: the molecules cluster and uncluster as the pressure wave moves through the gas. Note the molecules are not, on average, moving to the right; the impression that they are is an optical illusion. On average they are oscillating back and forth. Only the wave of pressure is, on average, moving to the right. Above: In green, a graph of the pressure field P(x,t) as a function of x, as it evolves in time. (M. Strassler 2012; All Rights Reserved)

Important Lessons

What do we learn from these four examples, all of which happen to exhibit Class 0 waves? (Recall that a Class 0 equation of motion has one quantity in it, called cw, and all waves in the corresponding field travel with speed cw. Different Class 0 fields have different values of cw.) We learn that the same field behavior can emerge from physically distinct fields in physically different media. Despite their different origins, the waves in a height field, in a lattice displacement field, in a magnet orientation field and in a gas pressure field may look identical as far as the field is concerned. They satisfy the same type of equation of motion and the same qualitative relationship between frequency and wavelength.

[Fine point: Strictly speaking, if you make the waves of sufficiently short wavelength, you will eventually be able to tell the difference between the different media. Once the wavelengths are as small as, say, the distance between the atoms in the rope, or crystal, or magnet, the wave equations that the waves satisfy will turn out to be more complicated than the ones we’ve written down, and the detailed difference will distinguish the media from one another. But often, in any practical experiment, we won’t come even vaguely close to seeing these effects.]

A consequence of this is that studying the waves (and their quanta) associated with a field does not uniquely tell you what the underlying medium is or what the physical interpretation of the field is — what property of the medium it represents. Or even if for some reason you happen know the field is of a certain type — a pressure field, say — you will still not in general be able to tell, from its behavior, what it represents the pressure of. All you can do by studying the waves is learn whether the field is of Class 0 or Class 1, and what cw is for that field is; or perhaps it is of some other class altogether.

In some ways that’s too bad; the field only carries partial information about the medium. In other senses that’s kind of neat: the field is a more abstract and universal thing than the physical material that it characterizes.

The field thus does not determine the medium; and its behavior is often independent of the details of the medium. Which then raises a question.

Can one have a physical field — with waves made from quanta that move through space and carry energy — without having a medium to support it?

A Field With No Medium?

A song cannot be heard without a singer. Yet a song has some kind of independent existence; it sounds different in detail, depending on who is singing it, yet there is something essential about the song, some abstract quality that makes it always recognizable. That abstract entity is the tune of the song. We can discuss and study and learn the tune, and write it down in musical notation, without ever hearing it sung by a singer. Many of us can even hum the tune silently in our heads. Somehow the tune exists, even if no one sings the song.

If, as in all the examples I’ve given above, and in all the others I could give you that would make intuitive sense to you, a field describes a property of a medium, then how can one have a field without a medium? Yet somehow fields are also independent of their media, because many different fields can behave in the same way, despite describing many different properties of vastly differing media. So perhaps it is possible to abstract the field away from its medium.

Well, not only is it possible, it is apparently mandatory. At least, it is mandatory either to have no medium at all, or to have a medium which can’t be made from ordinary matter, and is vastly different from all the media we’ve considered up to now — in that it acts as though (for all experiments anyone has ever done) it isn’t there.

One of several radical elements in Einstein’s 1905 theory of special relativity was the notion that for light waves — which were known for decades to be waves in electric and magnetic fields (“electro-magnetic waves”), all of which travel at the same speed in empty space — there is no medium. There are only the fields. [The hypothetical medium had been called the “aether”; Einstein argued there was no such thing, and wrote down a set of equations where indeed none was required. Note there is still debate (often more philosophy than physics) about whether one should or shouldn’t think of there as being a weird sort of medium that’s profoundly different from ordinary matter. At present there’s no experimental evidence that requires it. Please take any comments on this point to this webpage, rather than using the comment space below, which I would like to reserve for more straightforward physics questions, rather than controversies and debates.]

The key elements of Einstein’s version of relativity (as opposed to Galileo and Newton’s) were that

Space and time are not what you think; different observers, in steady motion relative to one another, disagree about the lengths of objects and the times between events (in a very precise and measurable fashion.)

There is a universal speed limit, called “c”; any observer, measuring the speed relative to him or her of any passing object, will find it to be less than or equal to c.

In such a world, certain fields — “relativistic fields” — can exist without a medium, and they satisfy special equations of motion. The simplest relativistic fields satisfy Class 0 or Class 1 equations of motion, with the wave speed cw that appears in the equation of motion taken equal to the universal speed limit c.

In short there are Class 0 relativistic fields that satisfy the equation

Class 0: d2Z/dt2 – c2 d2Z/dx2 = 0 .

(Light, and indeed all electro-magnetic waves, furnishes the most famous but not unique example; that’s why “c” is often called “the speed of light”.) And there are Class 1 relativistic fields satisfying the equation

Class 1: d2Z/dt2 – c2 d2Z/dx2 = – (2 π μ)2 (Z-Z0)

We’ll see examples in the next article; note there are no constraints from relativity on μ (except that μ2 be positive) or on Z0. [There are more complicated equations that are allowed for relativistic fields, but most of them, in simple physical processes, boil down to one of these two.]

Relativistic fields are physically real and physically meaningful in the universe, in that

their waves carry energy and information from one place to another

waves in one field can affect waves in another field and change physical processes that would happen in their absence

But unlike the fields in the examples given early in this article, relativistic fields do not describe a property of some ordinary physical medium that is made from anything resembling ordinary matter, and as far as is known experimentally, they do not describe the property of anything at all. These fields may be, as far as we know today, among the fundamental elements of the universe.

If you want to learn more about why an ordinary medium (like the rope, the lattice, the gas and the magnet in the examples above) is unacceptable for light waves, click here for an article comparing ordinary and relativistic fields and looking at how media and waves behave as viewed by different observers. [under construction]

If you want to join an ongoing discussion about whether it makes sense for a field without a medium to be a physical object (rather than just an abstraction), and what properties a medium suitable for a relativistic field would have to have (hint: it would have to be essentially identical to having nothing), then click here .

In the lattice displacement wave, what is waving back and forth is the position of each individual atom.

In the air pressure wave, the atoms are whizzing around at high speeds, and their individual positions and velocities do not undergo wavelike behavior. Only the collective measure of what the atoms are doing, the density and pressure, are undergoing wavelike oscillations, growing and shrinking regularly.

Similar, after the wave in the lattice displacement field passes, each individual atom returns to its position in the lattice. But in the air, atoms are moving around freely; so after the pressure wave passes, atoms do not return to their positions before the wave passed; they continue moving around freely. Only the pressure and density return to their initial value before the wave passed.

So they are actually pretty different — as different as solids are from gases.

That’s not the quote I see. I see: “However unlike the classical fields of Maxwell’s theory, QFT fields generally exist in superpositions of states.” The word “states” is important.

The problem here is that this statement in Wikipedia is not very clear.

To understand this we have to start with quantum mechanics of simple objects, not quantum field theory. In classical mechanics, in three dimensional space (with coordinates x,y,z), an object has, at any particular time, a definite position (X,Y,Z). We may say that the probability of finding the object at (X,Y,Z) is 1, and the probability of finding it at any other point (x,y,z). But in quantum mechanics, an object generally does not have a definite position; the probability of finding it at a position (x,y,z) is given by a wave function Ψ[(x,y,z)] (more precisely, by the square of that function.) Generally, the square of that function is not 1 at one point and 0 everywhere else; generally that function is non-zero over a large region. [Experts will see that I am slightly oversimplifying here — just slightly.]

What about in quantum field theory? In classical field theory, such as in Maxwell’s, a field is a function φ(x,y,z) of space (not to be confused with a wave function — we’ll see the wave function for field theory in a moment.). At any given time, the field has a definite value at every point in space. The probability that it has that value is 1; the probability that it has any other value is zero. In quantum field theory, however, the field does not have a definite value at points in space. Instead, it has a probability given by the square of a wave function to take any value at any point in space. The wave function is a function not of space but of all possible values that the field might take at all positions in space: Ψ[φ(x,y,z)]. So in classical field theory the field has a definite value at any point in space and time (just as in classical mechanics an object has a definite position at any moment in time) but in quantum field theory the field may generally take many possible values at any point in time, with probabilities given by a wave function of those possible values (just as in quantum mechanics an object may generally be at many different positions at any point in time, with probabilities given by a wave function of its possible positions.)

This is not what the Wikipedia article said; the wording is imprecise and in some ways misleading. But I do think this is what it was trying to say.

A field in physics is space filled with aether and the strength of the field is the displacement of the aether from its rest position.

“It is ironic that Einstein’s most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed [..] The word ‘ether’ has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum. . . . Relativity actually says nothing about the existence or nonexistence of matter pervading the universe, only that any such matter must have relativistic symmetry. [..] It turns out that such matter exists. About the time relativity was becoming accepted, studies of radioactivity began showing that the empty vacuum of space had spectroscopic structure similar to that of ordinary quantum solids and fluids. Subsequent studies with large particle accelerators have now led us to understand that space is more like a piece of window glass than ideal Newtonian emptiness. It is filled with ‘stuff’ that is normally transparent but can be made visible by hitting it sufficiently hard to knock out a part. The modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic ether. But we do not call it this because it is taboo.” – Robert B. Laughlin, Nobel Laureate in Physics, endowed chair in physics, Stanford University

Laughlin’s program has yet to be borne out in equations. There were proposals, and they didn’t work mathematically. I know Laughlin very well; he likes to make statements that are a little premature. I remember him insisting in my presence that his theory of high-temperature superconductivity was obviously correct (but it wasn’t.) Just because a person won a Nobel prize does not mean that everything they say after that is correct. For example, Einstein’s program to unify electromagnetism and gravity failed; he was using the wrong methods. That does not diminish his accomplishments.

and in case you are wondering: I was a Stanford graduate student and have known Laughlin for almost 25 years. I have heard him make many correct statements and many incorrect statements. He certainly knows what he is talking about, but he often states things as fact that are only conjecture; that’s his personal style.

In this instance, an argument from authority is not a valid logical method, as it does not represent the consensus of modern experts in the field. Please do not enter into our physics discussion with statements like this.

Einstein, for example, once told Lemaitre that his math was correct but his physics was “abominable” with regards to the Big Bang hypothesis. Great men can be wrong just like us mortals.

I’m sorry, but you do not know what you are talking about. Your statements about non-baryonic matter and aether are completely wrong. You are welcome to make incorrect statements on your own website, but please do not bother my readers, who are trying to understand physics.

You are misleading your readers when you state, “The hypothetical medium had been called the “aether”; Einstein argued there was no such thing, and wrote down a set of equations where indeed none was required.”

Einstein stated, “According to the general theory of relativity space with ether is unthinkable”.

Well, I’m happy to try to get the history correct. In his original papers on *special relativity*, he does exactly what I said. Please find me the precise citation of the statement that you’ve quoted above.

Many thanks for this series of articles, they’re great!
I have a question that maybe is a bit weird, but I have a problem trying to imagine a three dimensional wave…
Let’s see, in one dimension (X Axis), like a rope, a wave amplitude oscillates in the Y axis so to me it’s more a 2D wave. In two dimensions I imagine like a flat plane (XZ Axis) like ripples in a pond, but the amplitude oscillates in the Y (up) axis, so it looks like it needs three dimensions.
So… when I reach to the three dimensional wave, I imagine something like a spherical wave, but where it’s oscillating?? doesn’t need a 4th dimension to work? or maybe it’s just a value in the field but not really the medium what’s oscillating? Any example of a 3D wave?

“Let me emphasize one last time a point that often causes confusion. In general the field may have nothing to do with a physical distance in space. Yes, in articles 3 and 4 I used the example of a wave on a rope to illustrate what Z(x,t) might be, because it’s nice and intuitive; and also I’ve made graphs of Z(x,t) for waves many times. Both of those things might tend to give you the erroneous impression that Z(x,t) always has to do with waves that cause a physical object (like the rope) to move a distance Z in a direction of space perpendicular to the x direction. But that’s not true, as three of our four examples will illustrate.”

You are getting confused precisely because you’re thinking of Z as a direction in space; that’s why you’re thinking of the rope height wave as a 2D wave instead of as a 1D wave as you should. There is Z (the field) and there is x (space) — one dimension. If you think carefully about the other examples, you will see (especially in the case of the lattice) that there is no way that you should be thinking about two dimensions.

Only when you’ve convinced yourself that the ripples on a pond are a 2D wave, not a 3D wave, will you understand that the sound waves in a gas are a 3D wave.

Thanks Professor for your fast reply!
Ok, I think I understand now, so we have to think of Z as just a value of the field, isn’t it? a scalar, a vector or whatever but it’s just a measured value of the field at some point without space dimensional significance then.

Cool, thanks again for this series, you should write a book in that style 🙂

By the way, as some people were asking some days ago how to print this blog easily, I use this: http://pdfcrowd.com/
Converts any web to PDF format, so you can easily print it or read it on your ebook (like me) or whatever

I think Tony’s point involved how we should visualize a wave (if it didn’t, well, my question does 🙂

Your point with the range of examples seems to be that the same wave can arise from different media (or in fact no medium at all), so in trying to visualizing the wave, we should try to visualize the green line, and not the actual details of the medium.

So, your four examples of 1D waves all need THREE dimensions to visualize: the animated green lines have two spatial dimensions and one time dimension (for each position on the x axis and frame number N of the animation, there is plotted Z(x,N) on the y-axis).

In the rope example, the Z value actually coincides with the displacement of the medium, and you rightfully point out that in other types of 1D wave this doesn’t happen. However we still need that second spatial dimension in order to draw a graph of the wave.

What would an equivalent to this green line animation be, for the ripples on a pond? Well, we can cope with this example because it is just like your rope: the value Z(x,y,t) of the ripple amplitude coincides with the height of the water, and we’re used to visualizing three spatial dimensions (x, y, and Z) moving in time.

Now the confusing part comes: how do you graph a 3-d wave, F(x,y,z,t) ? In your mind’s eye there has to be an “extra” direction to put F, for every point in space (and everything is still changing in time, too).

Further on my own point, I got to thinking: what if you tried to draw F(x,t) , giving a spatial axis for ‘t’ (so the axes are x, t, and F)

If we take F = cos (x – t) to keep it simple then it would just look like a piece of corrugated iron, it is just like the ‘x’ and ‘t’ are not separate dimensions any more. (we can take one dimension out of this picture and still have the whole picture, i.e. just look at the two axes (x – t) and F. Is this what you’re getting at when you described F(x,t) as one-dimensional despite it being a function of two variables? Do all waves have this property (of course, general functions f(x,y) of two variables may not).

I feel there’s an important piece of understanding here that I am just skirting around the edges of , but not finding 🙂

Are you familiar with the old RKO Radio Picture logo? (If not, it’s easy enough to find online.)

The logo is concentric spheres — representing radio waves — emerging from the tip of a transmission tower.

Radio (and other E-M) waves are among the most common 3-D waves we’ll encounter in real life, along with sound, so it’s a pretty practical example. (The details of the logo — the speed, of course, but also the wavelength — aren’t accurate for real-life radio waves, though.)

many thanks, yes, that’s a good example. I think I understand now, my mistake was thinking of Z as a dimensional value instead of a value of the field at that point. So now I can picture a 3D sound wave as a sphere growing with “wavy” values of air pressure inside it 🙂

I found the easiest way to try to visualise waves in three dimensions to be to think of maximum value of the field as red, minimum value as purple and the rest of the rainbow in order in between. I think colour is the easiest property (other than length, positions and the like) to visualise.

Great article! It would add to the article in the introduction if you could go into the history a little more: who created the idea of a field? – Faraday. Why did they do it? – to account for action at a distance as mechanical stress in electromagnetic substance. Why did they call it a “field”? not sure… 😉

Medium is not energy. Medium can have some energy, but it is not the same. The oscillating value is the value of the field. For example you can have temperature field: you start with 7 degrees, a little later there is 8 degrees, later 9 deg, then back to 8, followed by 7, then 8, 9, 8, 7, 8, 9, 8, 7, …

The medium can be for example the air (or water, or gasoline, ethanol, or even something solid – like a rock, wood, brick). The oscillations are in the value of the field – the temperature. There is no value of medium: what is the value of rock?

Very nice description. Its funny, but I didn’t really get this sort of interpretation of particles and fields as described by physicists until rather late in the game. It really took a heavy dose of graduate level mathematics (fibre bundles, and then operator valued distributions) to finally go back three steps and accept part of this picture. And even now, having worked in the field for some time, I still find myself coming back and wondering whether we really know what we’re talking about. Quantum fields and their ‘particle like excitations’ are just so darned peculiar.

Thank you for doing this series, Matt. Intuition about these topics is really difficult to develop unless you’ve gone deep into the physics rabbit hole already. I was not really sure in what manner quantum fields were analogous to the wave PDEs I’m used to dealing with, so this is really helpful.

I was discussing mechanics with a guy taking his first college physics class, and I had the realization that my brain has been trained to think of certain physical ideas as “making sense.” Yet there is a point where my education runs out and I have no intuition at all. I try to get more people to read your site, since I think that is the most valuable service you are doing here: changing people’s perception from how the natural world “should” work to how it actually works.

As far as I can remember my college lessons from physics, a field is a physical concept that associates a certain value of a given physical magnitude to each and every coordinate of a metric space.

If the physical magnitude in question is a scalar (a scalar physical magnitude is any given magnitude that requires only one value to be completely determined, like, say, temperature, or preassure), it is a scalar field.

If the physical magnitude in question is a vector (a vector physical magnitude is any given magnitude that requires three values to be completely determined, magnitude, direction and sense like, say, speed, or gravity), it is a vector field.

Any heavy body, like a star or a planet, produces a gravitational field around it, so, for any given coordinate of space around the star or planet, there is a given value (a vector) of gravity associated with each and every position.

If I’m not mistaken, the Higgs Field is the first fundamental scalar field that has been theorized.

TO Martin : thanks , but you did not answer the basic question ; for the EMF or electron or quark fields what do you mean by “” value of the field “” ? what is the physical meaning of ( value of electron field ? )
You mentioned temperature field , but here we have air and a scale for air molecules behavior –that is temperature — , for the electron field clarify the 2 components so we can grasp the meaning .

If I’m not mistaken, the physical meaning of the quantum waves is the square root of probability density. So the “unit” is m^(-3/2). In other words: it is square root of inverse cubic meter. If you take the square of the quantum wave, you get the probability density: i.e. the probability that the particle will be detected in the given volume. For this reason it is probability per volume (thus the inverse cubic meter). If I’m not mistaken.

Martin –you are in fact **deeply** mistaken! A classic and common error. VERY IMPORTANT!

Do not confuse a FIELD — a function of **space** and time — with a PROBABILITY AMPLITUDE (often called the “wave function”) — a function of **all possible configurations of the system** and time.

The only situation in which a probability amplitude is a function of space and time, accidentally similar to a field, is when it describes a single point particle, which at any time t would have position (x,y,z). Even for two particles, with positions (x1,y1,z1) and (x2,y2,z2), a probability amplitude (or wave function) is a function of seven variables, Phi(x1,y1,z1,x2,y2,z2,t); for three particles it is a function of 10 variables, Phi(x1,y1,z1,x2,y2,z2,x3,y3,z3,t). For a field Z(x,t), the wave function is a functional! Phi[Z(x),t], a function of an infinite number of variables, telling us the probability amplitude at time t that the field takes a particular shape Z(x). Phi, the probability density, is NOT a field, and it is not in any way like a field. The square of the probability amplitude is a probability and is bounded, since probability has to be less than one; there is no constraint on the square of a field.

Could you please sketch a way how to relate a QFT field A(x,t) – that features localised ripples (particles) – to a probabilistic wave function presented in introductions into quantum mechanics? So, how do you calculate the probability that some apparatus would catch the photon at the place of the correspondent ripple in its field?

Do I need to describe my apparatus (e.g. silver nitrate) by another field B(x,t) that is excited at the same area where my photon ripple passes through? Then in QFT, there might not exist such a thing like an independent space-time probability density for photons. Rather you would always need to introduce other fields that interact with the photon, in order to tell where they will manifest themselves in space-time with a probability that is calculated from the yAB terms on the following pages. Is my thinking correct?

Also, I assume that the local field ripples that represent photons in the following pages do not really have one exact frequency/energy since they vanish eventually (Heisenberg). So I suspect that these local ripples in space time represent some composite photon state that is only approximate to a perfect one-photon as defined by energy steps. Is my confusion justified here?

My question was *not* about the wave function for the field strength that is the functional Ψ[φ(x,y,z)] in your comment above. I was wondering if there is such a thing like a probability density psi(x,y,z, …) in QFT to “find” a (one, multiple) photon(s) at a given spot (x,y,z) and how it is related to the ripples in the electromagnetic field. My suggestion was that this is not provided as a basic entity by the formalism of QFT but that it needs to be calculated point by point by introducing a the coupling to another field B(x,y,z) that represents the detection mechanism. But I’m just guessing here.

Anyway – the question about the physical meaning of these fields should be answered by this: “…relativistic fields do not describe a property of some ordinary physical medium that is made from anything resembling ordinary matter, and as far as is known experimentally, they do not describe the property of anything at all. These fields may be, as far as we know today, among the fundamental elements of the universe.”

If the particle is always detected entering, traveling through and exiting a single slit in a double slit experiment then how is this not evidence the particle always enters, travels through and exits a single slit and it is the wave in the field which passes through both?

Q. In a double slit experiment, why does ‘something else’ occur when the particle is not detected entering, traveling through and exiting a single slit?
A. ‘Something else’ does not occur. The particle always enters, travels through and exits a single slit. It is the associated wave in the field which passes through both.

Q2. Is there an energy quanta, (size, variables)? If one set t=very small duration in the uncertainty principle, deltaE x deltaT ~ h-bar is it valid to say h-bar is the small quantity of energy that can change state? i.e. lose it initial value for for ever and cannot go back to it so as not to violate the second law of thermodynamics?

Q3. The fact that we live in a four dimension universe say any thing with regards to the fundamental nature (state(s)) of energy?

Q4. How do they remove the neutrons from the nucleus in the particle collider?

Q5. Finally. is there a relationship between the answer of question 4 and the first 4 questions? i.e. Is understanding quantum confinement the Holy Grail in the universe?

Why are electrons (or any of the fermions), all exactly the same? Isn’t it an indication that a uniform grid, lattice, of a highly structured and uniformly packed energy spheres exist and permeate throughout the universe? The electrons and the other particles are the resonances and their harmonics of free energy moving in the cavities around these spheres. Their paths would be sinusoidal and the flow closer to the spheres would be more be fermion “massive” while the flow in between, the region of least resistance, would be bosonic, no mass. The mass in this model would be a property of the gravitational pull of the highly dense energy spheres. One lattice that defines everything. You may call these spheres gavitons but I say there is no need, since the “field’ is merely a manifestation of the geometry of the lattice held together by the extreme kinetic energy unleashed by the Big Bang.

This is why I asked Is quantum confinement the Holy Grail in the universe? The force that would be keeping the spheres intact would be enormous and so small and highly dense that would never be able to break apart.

Black holes could very well be a coalescing of these spheres into a bigger and bigger one. Yes, the question of why the universe expanding and not contracting could be asked, since the model would pull the galaxies together. Well is the universe really expanding? I can think of a construct that would have the galaxies moving apart within a contracting universe. Visualize a very thick shelled soap bubble, while the galaxies are floating away from each other in the shell the bubble is shrinking in size. To us the galaxies are moving away but spacetime is contracting.

Clearly the air pressure wave has no medium, just like light. The net wave length is carried by the random movement of the particles. The pressure field may contain multiple wavelength characteristics or harmonics. If the particles are bent by a gravitational field the wavelengths should remain unaffected. If the light strikes a medium where it is slowed down by a crystal lattice it will split into its characteristic harmonics. If it tries to go through a narrow slot a similar thing will happen based on net amplitudes of each net harmonic. Particles can even form into larger masses.

Strangely I find myself in sudden need of the promised article on “comparing ordinary and relativistic fields” – or a recommended substitute pro tem. Might I ask if you are in a position to offer either at present?

I’m afraid that what I am is without electrical power. I think what you want is a book on basic relativity that describes how Lorentz contractions in space and time are different from Newtonian space, and the doppler effect in non-relativistic and in relativistic fields, and how it is different.

Matt,
The song has an existence that is independent of the singer … that is one superb metaphor, as well as one heck of an effective pedigogical device. It reminds me of a phrase I use when I want to emphasize the distinction between an abstract object and any of its concrete representations: “Do not confuse the person for his clothes.”
Keep up the good work, Matt. Your expository skills continue to amaze and captivate.

I don’t understand electromagnetic fields. In quantum mechanics forces (or interactions) are due to exchange of bosons between particles. In Perturbation Theory they exchange virtual bosons when near each other. How two distant electrons (say at one millimetre of distance) interact between themselves through exchanging virtual photons? I mean two electrons interact in complete darkness too.

I’m reading your article and am commenting on Fig 2. To understand it, I copied a snapshot of the animation and add the image of the lattice at equilibrium underneath. This way I can see the dots displaced from equilibrium. Drawing a line from the equilibrium dot to the displaced counterpart, I can “see” that a positive D(x,t) is due to the dot displaced forward and a negative D(x,t) is a dot displaced backward. I have an image of this construction, but can’t find a way to give it to you. Thank you – this is all more understandable than any other course’s explanations. Most important are the disclaimers so that I don’t take an analogy as a fact and get confused.